Improved Facility Location for Logistic Network Design ... · Shaju Varghese 1 2and Gladston Raj S...

Global Journal of Mathematical Sciences: Theory and Practical.

ISSN 0974-3200 Volume 9, Number 1 (2017), pp. 65-79

© International Research Publication House

http://www.irphouse.com

Improved Facility Location for Logistic Network

Design using the proposed K-means - GA Fusion

Model

Shaju Varghese 1 and Gladston Raj S 2

1 Research Scholar, Bharathiar University, Coimbatore, Tamilnadu, India. 2Head of Department of CS, Govt. College, Nedumangadu , Thiruvananthapuram,

Kerala, India.

Abstract

Logistic Network design and management is the process of identifying and

locating an optimal location for providing the facility. The location or the

decision of where to locate the facility is important and it determines the

efficiency and effectiveness of the logistic management. The distribution

centers(DCs) should located in easy to access road ways and closer to the

populated area to accelerate the delivery and also to reduce the overall

transportation cost and corresponding time. Also, it is a tedious process to find

the optimal location for a DC and to minimize the number of DCs and thus the

overall cost.

In our previous work[19][20][21], we proposed the soft computing based

models for facility location in logistic network design. In our previous

work[19], we presented a genetic algorithm based model for facility location.

In this work we propose a K-Means – GA fusion hybrid model for facility

location in logistic network design.

Keywords: Logistic, Heuristic, Hybrid, Inbounded, Crossover, Mutation,

Reproduction, Population, clustering, Optimal

I. INTRODUCTION

The basic concept of facility location analysis in logistic management is the K-center

problem and P-median model. It is the process of identifying the facility location and

generates a common model for finding the optimal locations. In this work it evaluated

by two soft computing models known as k-means clustering and genetic algorithm.

66 Shaju Varghese and Gladston Raj S

Also, these model generate a hybrid model using these models to locate an optimal

locations for facilities.

Facility Location Problem(FLP)

The facility location analysis and identification is a challenging and non-linear

problem in the areas of resource procurement & management, production, supply,

warehouse management. This problem is faced by many organizations in their

operational stages. The requirement of location decision-making has led to the

development of the location analysis and modeling as a part of the operations

research[3]. This is the process of finding the optimal locations (one or more) for this

problem in order to locate minimum service distance for maximum number of users.

The facility can be identified as the bases, units, equipment, weapon systems,

logistics, civil objects, etc.[3]. Various levels of researches have been conducted in

this area and they proposed various approaches as the solution.

Location models are often difficult to solve, especially for large problem instances[3].

There are number of built in commercial tools are available that can solve the

complexity of a location models. Besides location models are application dependent.

Their objectives, constraints and variables are determined by particular problem under

study. It is very difficult to develop an all-purpose model that can solve the common

facility location problem that is optimal for existing applications[3].

The major solution techniques are

Exact Solution Techniques.

Heuristic Solution Techniques.

The heuristics solution methods are

Bound on Optimal Solutions

Worst Case Analysis

Statistical Evaluation

Applications of FLP

The basic mathematical modeling concept and algorithms that are suitable for

developing this type of mathematical models are the basic goal of this type of

development. But the areas of innovative applications of this type of model are also

important and also are considered at the time of development. The common

applications include land use optimization for emergency facilities, transport route

identification and optimization, retail shop outlet location selection, spatial objects

optimization, performance optimization of urban modeling. When applied to

telecommunication network design and capacity planning, the difficult task of

designing and maintaining huge, reasonably optimal, telecommunication networks

becomes more feasible. Many major telecommunication service providers are finding

it imperative to upgrade or expand their facilities and services. The facility location

models can make use of the design of various types of telecommunication networks.

The capacitated optimization technique can play an important role for the

Improved Facility Location for Logistic Network Design using the proposed K-means.. 67

improvement of the telecommunication network design. In this example, most of the

analysis decisions have to be made concerning relation between concentrator location

and cable expansion.

II. FACILITY LOCATIONS & LOGISTICS

The process of locating an optimal location for facilities with considering the

attributes like facility construction costs, transportation costs, etc. This issue will

affect the smooth transferring of the materials or products from one place to another in

normal or emergency situations. Many studies and researches in this area contribute

number solutions but most of the solutions are not common and purely concentrated

on a particular type of problem.

The purpose of this work is to build a decision support system that helps the managers

decide where to locate a facility.

Time complexity of problems of the above mentioned problem is a obvious one, so

that linear programming based methods will take much time with respect to the depth

of the problem and the assigned conditions. In this work, we will design a soft

computing model for solving a simple facility location problem with minimum

constraints.

Logistic Management

Logistics is the management of a transfer of things from one place to another.

Transportation is the mode of transfer in this process. This process involves the pack

movement, warehousing, packing, production, material handling, integration of

information flow, security etc.

History of Logistics

The concept of logistics was introduced late 19th century. Another definition for

logistics is that it is the detailed co-ordination of a complex operation consisting of

many people, warehouses, distribution etc. In a logistic management process the

planning, implementing and controlling procedures for the systematic and time

bounded transfer from one place to another.

The logistics activities can be classified as

1. Inbounded logistics – Arranging the internal movement of the materials.

2. Out bounded logistics – The products are moving from the production unit to

the end user.

The major access fields are procurement, distribution, after sales, disposal,

global/domestic, emergency, production, construction, asset control, reverse logistics.

Applications

1. Military Logistics


In this case it is very important to maintain the security. Also, the logistics

management failures cause critical problems. An Integrated Logistics Support

(ILS) is a discipline tool used to maintain the best service with lowest cost.

2. Business Logistics

The major goal of the business logistics is that distribution of the products in

the actual quality at the exact time at the correct place for the right price in the

perfect condition. Since 1960s only the concept of business logistics came to

the business environment. This is the basic concept of the shipping process of

worldwide supply chain management.

Facility Location in Logistics Network

In Bansal [2], a Public Logistics Network (PLN) for the continental U.S. was

designed. This research uses the same design approach that was developed by Bansal.

Bansal's design can be explained as four step process that includes generation of the

Underlying Road Network (URN), developing the network of public Distribution

centers (DCs), estimation of average package delivery time, and finding public DC

locations that minimize average package delivery time.

In this work, the PLN will be designed using a similar process with some

modifications and simplification in the steps. In this work we use a simplified version

of that design so that, instead of using the “average package delivery time” as a metric

for optimization, we used simple distance as the metric in the fitness function of the

soft computing model. This approach was used to minimize the optimization time.

Since the distance is directly proportional to package delivery time, we believe that

this approach also will lead to equal results, logically with in lesser time.

Generation of the Underlying Road Network (URN)

Developing the network of Distribution centers (DCs),

Finding public DC locations that minimize the distance between the DCs and the

User locations.

Figure 1: US states Map and their names


Generation of the Road Network of USA

The following map shows the road network that we created from the US census data

set.

Figure 2. Road Network of USA

Generation of the Underlying Road Network of Regional Distribution Centers.

The population in RDC is represented by total 925 U.S. census blocks that are plotted

on the map of RDC. A sub-graph of the road network was generated that is then

followed by the removal of two-degree nodes from the network. Each point in this

network is a potential location for a DC

The following graph/map shows the road network of Alabama(AL), USA that will be

the example of a sub graph we created and used to create the regional distribution

centers (RDC) that we are interested in.

Figure 3. Underlying Road Network of RDCs


Network of DCs

If needed, we may also create the network of DCs as follows.

The arcs connecting census blocks to the URN are added to the network and the

shortest time paths and distances between each pair of points are calculated using

Dijkstra's algorithm.. DCs will be located at some of the key points and then

connected to each other using Delaunay Triangulation [22] to form a network of

public DCs. The shortest time paths between all pairs of DCs is found and those paths

and distances are then used to calculate the percent flow of the packages, wij from

DCi to DCj using order based proximity factors developed by Kay and Parlikad [4]

Figure 4: Randomly placed facilities

K-Means Clustering

Simply put, k-Means Clustering is an algorithm among several that attempt to find

groups in the data. In pseudo code, it follow this procedure:

The vector m contains a reference to the sample mean of each cluster. x refers to each

of our examples, and b contains our "estimated class labels.

Explained perhaps more simply in words, the algorithm roughly follows this

approach:

1) Choose some manner in which to initialize the mi to be the mean of each group (or

cluster), and do it.

2) For each example in your set, assign it to the closest group (represented by mi).

3) For each mi, recalculate it based on the examples that are currently assigned to it.

4) Repeat steps 2-3 until mi converge.


Now that we have some rudimentary understanding of what k-means

Genetic Algorithm

Genetic algorithm is a population-based search method. Genetic algorithms are

acknowledged as good solvers for tough problems. It is an iterative procedure

maintaining a population of structures that are candidate solutions to specific domain

challenges. During each temporal increment (called a generation), the structures in

the current population are rated for their effectiveness as domain solutions, and on

the basis of these evaluations, a new population of candidate solutions is formed

using specific genetic operators such as reproduction, crossover, and mutation.

Figure 5: Optimum Locations by Genetic Algorithm

Initialize mi, i = 1,…,k, for example, to k random xt

Repeat

For all xt in X

bit 1 if || x

t - mi || = minj || x

t - mj ||

bit 0 otherwise

For all mi, i = 1,…,k

mi sum over t (bit x

t) / sum over t (bi

t )

Until mi converge


The K-Means-GA Fusion Method

We used linear Fusion approach in design. In this algorithm, instead of initializing the

population with random candidate locations, we first use the k-means algorithm for

doing the initial estimate on better starting locations.

The following pseudo code outlines the design of K-Means-GA Fusion for improving

Facility Location.

Function k-mean-GA_FLP

begin

kmeans : initialize random centroids

run k-means for n iterations to find

N optimum initial candidate locations

INITIALIZE population with N candidate

solutions provided by k-means algorithm;

(Each candidate solution will represent N

locations of the facilities)

EVALUATE each candidate;

(find fitness of each candidate using the

fitness function)

repeat

SELECT parents;

(Select two candidate having best

fitness value)

RECOMBINE pairs of parents;

(use single point crossover on the

selected candidates and generate new

population – this includes the original

parents)

MUTATE the resulting children;

(use gaussian mutation on entire

population)

EVALUATE children;

(find fitness of all new candidates of the

population)

until TERMINATION-CONDITION is

satisfied

end

Place the facilities on the locations optimized

by GA


Figure 6: Facility Location by k-means-GA Fusion Method

The Fitness Function

The following function is used to find the fitness at the set of facility location XYF=

(Xi, Yj ), where i,j = 1 to n. The set of points which has the lowest fitness vaue will be

the optimum location for placing facilities. XYU is the locations of all the

customers(cities, towns, villages)

Function d= EuclideanDist(XYU, XYF)

Begin

// Compute the Euclidean distance with each

coordinate

[R,C]=size(XYU);

//sum squared data - save re-calculating

repeatedly later

XYsq=repmat(sum(XYU.^2,2),1,NoDCs);

// The distance Function d^2 = (x-c)^2 = x^2

+ c^2 -2xc

Dist = XYsq + repmat (sum ( (XYF.^2)' ,1),

R, 1) - 2.*(

XYU *(

XYU '));

//label points

[d,Classes]=min(Dist,[],2);

d =sqrt(sum(d));

return (d)

End

The Fitness Function


III. RESULTS AND DISCUSSION

We have implemented the proposed soft computing based models for facility location

in logistics analysis using Mat lab software version R2012s. We used some of the

functions of Logistics Engineering Toolbox “Mat log Version 16” in this research. We

tried to use almost equal input parameters for each and every evaluated method. We

used the USA census data and map data which is much suitable for this kind of

research. We decided to use USA data because, it is the only data refereed in some of

the previous works and there seems no such detailed data available for any other

country for validating the methods of facility location and logistics analysis.

The Parameters of the Soft Computing Models

Genetic Algorithm options

StallGenLimit : 20

Generations : 200

PopulationSize : 20

CrossoverFcn : Two Point Crossover

MutationFcn : Gaussian k-means Clusterign

MaxRepetitions : 10

MaxIter : 20

The Time Taken for Finding Optimum Locations of DCs : 2.81 sec

The Average Distance between DCs and Customer Locations: 2.53Units

Figure : 7 - Genetic Algorithm Working Model


The Time Taken for Finding Optimum Locations of DCs : 1.57 sec

The Average Distance between DCs and Customer Locations: 2.39 Units

Figure 8: K-Means – GA Fusion Working Model

The following table shows the overall results of this work. Since the performance of a

soft computing model will depend up on several factors, and some random conditions,

we run each algorithm several times and only selected the values which are minimum.

Table 1: The Overall Performance for Locating 25 Facilities

Facility Location Method Avg. Distance Time Consumed

Random 6.16 -

GA 2.53 2.81

k-means-GA Fusion 2.39 1.57

The following graph shows the performance of the algorithms in terms of the

minimum average distance achieved. The average distance is the average of distance

between all DCs and the Customer locations. Each Customer is bound with a nearest

DC and the distance between each customer to that corresponding DC is calculated

and then the averages of all such distances were calculated. In this graph, the first

column shows the reference distance which is nothing but the initial average distance

of DCs that are randomly placed on the map.

With respect to the average distance, the k-means-GA fusion model performed well. It

means, k-means-GA found the optimum facility locations better than the Normal GA

methods.


Avg.Distance Achieved by Different Methods

6.16

2.53 2.39

0

1

2

3

4

5

6

7

Random GA k-means-GA Fusion

Facility Location Algorithm

Dis

tan

ce

in

La

t-L

on

g U

nit

s

.

Figure 9. The Performance in Terms of distance

The following graph shows the performance in terms of cpu time. Unexpectedly, the

k-means-GA model consumed less time. Even though the GA parameters were same,

the fusion model consumed lesser time. Even there was no increase in time for the

initial k-means step. Since the k-means step already found a good initial starting

location, the search process of GA becomes very focussed so that, the individual steps

of GA converge very fast. This may be the main reason for this low runtime of GA.

Time Taken for Facility Location

2.81

1.57

0

0.5

1

1.5

2

2.5

3

GA k-means-GA Fusion

Facility Location Method

Tim

e (

se

c)

.

Figure 10. The Performance in Terms of CPU Time

IV. CONCLUSION

Facility location for logistics is a wide area for research. In this work we addressed the

possibilities of using soft computing based models for facility location for logistics.

We used soft computing based clustering approach for Facility Location Problem &

Logistic Analysis.

As per the results, the hybrid, soft computing based optimization models successfully

found optimum locations of facilities in considerably meaningful time limit.

In this work, we used a simple Euclidean distance function as a fitness function in the

design of soft computing based location optimization model. Also the attributes like


distance, road types, and city population are included in the model. But, there are

much more constraints and parameters in a practical logistics problem that can be

included in the design of the fitness function such as (1)loading unloading time at

DCs, (2) different modes of travel times such as air travel time. This kind of more

constraints and parameters can be included in future design of soft computing based

optimization models. Our future works will address these issues.

We have designed the proposed models as minimum number objective problem. But

there are facility location and logistics situations where there may be more objective

during optimization. Future works may address the design of soft computing based

optimization models for multi objective optimization scenarios.

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[19] Shaju Varghese, Gladston Raj S, “A Genetic Algorithm Based Optimization

Model for Facility Location in Logistic Network Design”, International

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ISSN 0973-4562.

[20] Shaju Varghese, Gladston Raj S, “Simulated Annealing and Direct Search

Based Optimization Models for Facility Location in Logistic Network Design”

International Journal of Computer Applications, Vol 132, pp- 31-37,

December 2015, ISSN: 0975-8887.

[21] Shaju Varghese, Gladston Raj S, “ Clustering Based Model For Facility

Location In Logistic Network Using K-Means”, International Journal of

Scientific Inventions and Innovations, Volume 1 Issue 1, July 2016,


AUTHORS PROFILE

Mr. Shaju Varghese received his M.Sc. (Maths), M.C.A., and M.Phil.

in computer Science. Now working as Head of the Department of

Computer Applications at Baselios Poulose II Catholicos (B. P. C )

College, Piravom, Kerala, India. He was the Principal Investigator of

the Minor Research Project "Computerized Facility Location Analysis

In Rural Area Using Clustering", 2010, funded by Universities Grant

Commission, India. His research interest includes Data Mining,

Facility Location Problem, and Cyber Criminology, and has presented papers in

International and National Seminars. He has got three international journal

publications. Currently he is pursuing Ph.D. in Computer Science at Bharathiar

University, Tamilnadu, India.

Dr. Gladston Raj S. received his M.Sc (CS), M.Tech (Image

Computing) and PhD in Computer Science from University of

Kerala and Completed UGC-NET from University of Kerala and

PGDCH (Computer hardware) from MicroCode, He is Now

working as Head of the Department of Computer Science at Govt.

College Nedumangad, Kerala, India. His area of interest includes

Image Processing, Signal Processing, Data mining. He is providing research guidance

for Ph.D scholars from different areas of research and has presented several invited

talks in this areas of research.


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